EXPERIMENT 3

OBJECTIVE:
To study the role of different parts of universal vibration system and develop an understanding of
free and forced & damped and un-damped vibrations

APPARATUS:

Universal Vibration System

THEORY:

A vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium/mean

position. It can be useful as well as harmful, so we need to understand this phenomenon quite clearly
in order to avoid it or take advantage from it.

It can cause one or more of the following undesirable effects:

 Structural failure due to excessive displacement and stress

 Fatigue failure
 Noise generation
 Slippage and dislocation of joints
 Wear due to relative motion between components
 Discomfort while transportation
 Vibration transmission to connected structures
It can be used for positive purposes such as:

 Transporting powder
 Sieving and sizing products
 Consolidating the poured concrete
 Applying massage for muscle relaxation
 Removing dental plaque

A vibratory system consists of three basic parts: spring/stiffness element (stores potential energy),
mass/inertia element (stores kinetic-energy) and a damper (dissipates energy).

Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to
vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting go,
or hitting a tuning fork and letting it ring.

Forced vibration occurs when a time-varying disturbance (load, displacement or velocity) is applied to
a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, or a
random input. Examples of these types of vibration include a washing machine shaking due to an
imbalance, transportation vibration caused by an engine or uneven road, or the vibration of a building
during an earthquake.

Damped vibrations occur when the energy of a vibrating system is gradually dissipated by friction
and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change
in frequency or intensity or cease and the system rests in its equilibrium position.

Un-damped vibrations have no resistive force to act on the vibrating object. As the object oscillates,
the energy in the object is continuously transformed from kinetic energy to potential energy and back
again, and the sum of kinetic and potential energy remains a constant value.

Main Parts of Universal Vibration System:

1. Unbalance exciter 2. Beam 3. Damper 4. Exciter Control Unit
5. Drum for Recording Vibrations 6. Suspension and Oscillating Spring 7. Fr
Frame:
The experimentation set-ups are mounted in an aluminum sectional frame which is rigid but light
weight. Rapid attachment and simple adjustment of components using T-slots, T-slot blocks and
clamping levers can be attained.

Imbalance Exciter:
It is a kind of exciter in which an eccentric mass is provided which rotates and provides a forcing
factor to impart the forced vibrations.

Beam:
A cantilever is a beam anchored at only one end. The beam carries the load to the support where it is
forced against by a moment and shear stress.

Damper:
An oil filled damper is provided in the system for the dissipation of the energy produced during
vibrations via viscous damping. Its damping can be changed screwing the needle valve in or out.

Exciter Control Unit:

The unit controls the frequency of excitation through potentiometer and a digital counter is also
placed on it for displaying excitation frequency in Hertz. Power and recorder switches are also
provided on the unit.

Drum Recorder:
Drum recorder is placed in front of the stylus. Graph paper rotates at its outer surface and records
t he resulting response

Springs:
The helical suspension springs are available with universal vibration system to provide the required
stiffness.

Spring Characteristics:

D = diameter of coil, d=diameter of spring wire, k = stiffness

(a) 0.75N/mm , D= 18.3mm , d =1.05mm

(b) 1.5N/mm , D= 17.6mm , d =1.6mm

(c) 3.0N/mm , D= 16.5mm , d =2.1mm

Discussion
 EXPERIMENT 4

OBJECTIVE:
To determine the time period and frequency of free un-damped vibrations of a spring-dashpot system
and compare the experimental results with theoretical outcomes.

APPARATUS:
Universal Vibration System

THEORY:

A system is said to be a cantilever beam system if one end of the system is rigidly fixed to a support
and the other end is free to move.
Vibration analysis of a cantilever beam system is important as it can explain and help to analyze a number
of real life systems. Real systems can be simplified to a cantilever beam, thereby helping us make design changes
accordingly for an efficient system.

When given an excitation and left to vibrate on its own, the frequency at which a cantilever beam will oscillate
is its natural frequency, this condition is called free vibration. The value of natural frequency depends only on
system parameters i.e. mass and stiffness. When a real system is approximated to a simple cantilever beam,
some assumptions are made for modelling and analysis (Important assumptions for un-damped system are
given below):

 The mass (m) of the whole system is considered to be lumped at the free end of the beam
 No energy consuming element (damping) is present in the system i.e. un-damped vibration
 The complex cross section and type of material of the real system has been simplified to a
cantilever beam

The governing equation for such a system (spring mass system without damping under free
vibration) is as below:

𝑚𝑥̈ + 𝑘𝑥̈ = 0

𝑥̈ + 𝜔n2𝑥̈ = 0

“K” the stiffness of the system is a property which depends on the length (l), moment of inertia (I) and
Young's Modulus (E) of the material of the beam and for a cantilever beam is given by:

3𝐸𝐼
𝑘=
𝐿3
OBSERVATIONS AND CALCULATIONS:

Time period is the time needed for one complete cycle of vibration to pass a given point.
Frequency and time Period are in a reciprocal relationship. Theoretical time period & frequency are
calculated by following formulae

𝑚𝐿2 1
𝜏𝑡ℎ𝑒𝑜 = 2𝜋√ ; 𝑓𝑡ℎ𝑒𝑜 =
3𝑘𝑎2 𝜏𝑡ℎ𝑒𝑜

Where,

m= mass of the beam = 1.64kg L= length of the beam = 670mm

k = stiffness of the spring in N/m a = distance between beam’s fixed end and spring pivot point

Experimental time period & frequency are calculated by following formulae:

𝑑 1
𝜏𝑒𝑥𝑝 = ; 𝑓𝑒𝑥𝑝 =
𝑣 𝜏𝑒𝑥𝑝
Where;

d = wave length of vibration (mm) v = velocity of recorder = 20 mm/s

PROCEDURE:

1) Measure the length of the beam.

2) Set the distance ‘a’ from fix end of beam to spring pivot point and measure it.
3) Insert the stylus into the holder at the free end of the beam.
4) Wrap graph paper onto the recorder.
5) Dismount the unbalance exciter.
6) Switch on the apparatus.
7) Switch on the recorder from the control unit.
8) Now apply force by hand on the free of the beam in downward direction and release it.
9) The oscillations are started. Now wait for the beam to return to its initial position.
10) Switch off both the recorder and the apparatus.
11) Take out the graph paper from the recorder carefully.
12) Measure the distance between initial and end points of a single wave
13) Now calculate the time period and frequency using the above given formulae.
14) Repeat the experiment by varying the spring constant (k) and distance (a).
OBSERVATIONS AND CALCULATIONS

Spring Stiffness Distance (a) 𝜏theo. in 𝜏exp. in 𝑓theo. in 𝑓exp. in % error

Sr. No.
(k) in N/mm in mm sec sec Hz Hz

1 1.5 400

2 1.5 500

3 1.5 600

4 3.0 400

5 3.0 500

6 3.0 600

PRECAUTIONS:

1. Do not disturb the beam during vibrations.

2. Secure the stylus in the holder tightly so that correct pattern can be recorded.
3. Control unit should be operated with care.
4. Beam should be aligned horizontally before starting the experiment.
5. Base of the apparatus should be stable while taking measurements.

RESULTS AND DISCUSSION